Calculate the Laplacian of the following functions:

(a)

(b)

(c) .

(d)

The divergence of gradient is called as Laplacian of a vector.The vector is defined as. theoperator is defined as. The value of Laplacianis obtained as

Here,is the second ordered partial derivative of function.

(a)

To compute an expression substitute the vectors and other required expression and then simplify

The vectoris defined as. The Laplacian of the vector ois defined as.

Substitutefor into.

Thus the value of is 2.

(b)

To compute an expression substitute the vectors and other required expression and then simplify.

The vectoris defined as. The Laplacian of the vectoris defined as.

Substitutefor into .

Solve further as,

Thus, the value ofis.

(c)

To compute an expression substitute the vectors and other required expression and then simplify.

The vectoris defined as. The Laplacian of the vectoris defined as.

Substitutefor into.

Solve further as,

Thus, the value ofis 0.

(d)

To compute an expression substitute the vectors and other required expression and then simplify.

The vectoris defined as. The Laplacian of the vectoris defined as.

Substitutefor into .

Solve further as,

Thus the value ofis.